Overview of OpenMX

Taisuke Ozaki ISSP, Univ. of Tokyo

The 3rd OpenMX/QMAS workshop at ISSP, 11th-13th May 2015

• Introduction • Implementation of OpenMX • Ongoing and planned developments

About the slides

http://www.openmx-square.org/workshop/workshop15/OpenMX-Overview.pdf

The PDF file can be downloaded from the website for the program of the workshop.

Purpose of the tutorial

First-principles calculations based on the density functional theories (DFT) have been now widely accepted as a useful tool to understand chemical and physical properties of molecules and bulks with reasonable accuracy in relatively small computational cost.

OpenMX (Open source package for Material eXplorer) is a software package for nano-scale material simulations based on DFT, norm-conserving pseudopotentials, and pseudo-atomic localized basis functions. The methods and algorithms used in OpenMX and their implementation are carefully designed for the realization of large-scale ab initio electronic structure calculations on parallel computers.

In the tutorial, we aim to hand on the theoretical background of OpenMX for a wide variety of researchers who pursue deep understanding of materials properties from first-principles.

• Software package for density functional calculations of molecules and bulks

• Norm-conserving pseudopotentials (PPs)

• Variationally optimized numerical atomic basis functions

• SCF calc. by LDA, GGA, DFT+U

• Total energy and forces on atoms

• Band dispersion and density of states

• Geometry optimization by BFGS, RF, EF

• Charge analysis by Mullken, Voronoi, ESP

• Molecular dynamics with NEV and NVT ensembles

• Charge doping

• Fermi surface

• Analysis of charge, spin, potentials by cube files

• Database of optimized PPs and basis funcitons

• O(N) and low-order scaling diagonalization

• Non-collinear DFT for non-collinear magnetism

• Spin-orbit coupling included self-consistently

• Electronic transport by non-equilibrium Green function

• Electronic polarization by the Berry phase formalism

• Maximally localized Wannier functions

• Effective screening medium method for biased system

• Reaction path search by the NEB method

• Band unfolding method

• STM image by the Tersoff-Hamann method

• etc.

OpenMX Open source package for Material eXplorer

Basic functionalities Extensions

2000 Start of development 2003 Public release (GNU-GPL) 2003 Collaboration: AIST, NIMS, SNU KAIST, JAIST, Kanazawa Univ. CAS, UAM NISSAN, Fujitsu Labs. etc. 2013 17 public releases Latest version: 3.7

http://www.openmx-square.org

History of OpenMX

About 280 papers published using OpenMX

• T. Ozaki (U.Tokyo) • H. Kino (NIMS) • J. Yu (SNU) • M. J. Han (KAIST) • M. Ohfuti (Fujitsu) • F. Ishii (Kanazawa Univ.) • T. Ohwaki (Nissan) • H. Weng (CAS) • M. Toyoda (Osaka Univ.) • H. Kim (SNU) • P. Pou (UAM)

• T. V. Truong Duy (U.Tokyo) • C.-C. Lee (JAIST) • Y. Okuno (Fuji FILM) • Yang Xiao (NUAA) • Y. Gohda (TIT)

Developers of OpenMX

Current status of OpenMX

The code with 0.3 million lines is written by a standard C, and partially Fortran90 including 1 makefile, 21 header files, 261 c files, 4 fortran90 files.

The code is publicly available under GNU-GPL.

Manual in English and Japanese, and technical notes on implementations in English.

Database of optimized pseudopotentials and basis functions for 76 elements.

OpenMP/MPI hybrid parallelization. The parallel efficiency of 68% is achieved using a hundred and thirty thousands cores on K.

Publications with OpenMX

First characterization of silicene on ZrB2 in collaboration with experimental groups

B.J. Kim et al., Phys. Rev. Lett. 101, 076402 (2008). First identification of Jeff=1/2 Mott state of Ir oxides

A. Fleurence et al., Phys. Rev. Lett. 108, 245501 (2012).

K. Nishio et al., Phys. Rev. Lett. 340, 155502 (2013). Universality of medium range ordered structure in amorphous metal oxides

C.-H. Kim et al., Phys. Rev. Lett. 108, 106401 (2012). H. Weng et al., Phy. Rev. X 4, 011002 (2014).

Theoretical proposal of topological insulators

H. Jippo et al., Appl. Phys. Express 7, 025101 (2014). M. Ohfuchi et al., Appl. Phys. Express 4, 095101 (2011).

Electronic transport of graphene nanoribbon on surface oxidized Si

H. Sawada et al., Modelling Simul. Mater. Sci. Eng. 21, 045012 (2013).

Interface structures of carbide precipitate in bcc-Fe

T. Ohwaki et al., J. Chem. Phys. 136, 134101 (2012). T. Ohwaki et al., J. Chem. Phys. 140, 244105 (2014).

First-principles molecular dynamics simulations for Li ion battery

About 300 published papers

Z. Torbatian et al., Appl. Phys. Lett. 104, 242403 (2014). I. Kitagawa et al., Phys. Rev. B 81, 214408 (2010).

Magnetic anisotropy energy of magnets

Silicene, graphene Carbon nanotubes Transition metal oxides Topological insulators Intermetallic compounds Molecular magnets Rare earth magnets Lithium ion related materials Structural materials etc.

Materials treated so far

Materials studied by OpenMX

Implementation of OpenMX

• Density functional theory • Mathematical structure of KS eq. • LCPAO method • Total energy • Pseudopotentials • Basis functions

Density functional theory

W.Kohn (1923-)

The energy of non-degenerate ground state can be expressed by a functional of electron density. (Hohenberg and Kohn, 1964)

The many body problem of the ground state can be reduced to an one-particle problem with an effective potential. (Kohn-Sham, 1965)

1. Band gap: Underestimation of 30～40 % 2. vdW interaction: No binding in many cases 3. Strongly correlation: No orbital polarization of localized d- and f-states

Atomization energy: 0.3 eV (mostly overbinding) Bond length: Overestimation of 1 % Bulk modulus: Underestimation of 5 % Energy barrier: Underestimation of 30 %

Mean absolute error

Successes and failures of GGA

Successes: 1. Accuracy:

2. Accurate description of hydrogen bonding 3. Better description of magnetic ground states (e.g., bcc Fe)

Failures:

3D coupled non-linear differential equations have to be solved self-consistently.

Mathematical structure of KS eq.

Input charge = Output charge → Self-consistent condition

OpenMX: LCPAO

OpenMX: PW-FFT

2000 block BOP PRB 59, 16061, PRB 61, 7972

2001 ab initio recursion method PRB 64 195126

2001 O(N) inversion method by recursion PRB 64, 195110

2003 Variational optimized basis functions PRB 67, 155108, PRB 69, 195113, JCP 121, 10879

2005 Efficient implementation of LCPAO method PRB 72, 045121

2005 O(N) LDA+U method PRB 73, 045110

2006 O(N) Krylov subspace method PRB 74, 245101

2007 Efficient integration of Green’s function PRB 75, 035123

2008 Non-collinear DFT with constratints web notes

2009 First evaluation of two-electron repulsion integrals JCP 130, 124114

2009 First spherical Bessel transform method CPC 181, 277

2010 Non- equilibrium Green’s function method PRB 81, 035116

2011 O(N2~) method PRB 82, 075131

2011 O(N) nearly exact exchange functional PRA 83, 032515

2012 O(N) + ESM method JCP 136, 134101

2014 Massively parallelization of OpenMX CPC 185, 777

2014 Massively parallelization of 3D-FFT CPC 185, 153

2014 Large-scale NBO analysis JCP 140, 244105

Papers on methods in OpenMX

One-particle KS orbital

is expressed by a linear combination of atomic like orbitals in the method.

Features: • It is easy to interpret physical and chemical meanings, since the KS

orbitals are expressed by the atomic like basis functions. • It gives rapid convergent results with respect to basis functions due to

physical origin. (however, it is not a complete basis set, leading to difficulty in getting full convergence.)

• The memory and computational effort for calculation of matrix elements are O(N).

• It well matches the idea of linear scaling methods.

LCPAO method (Linear-Combination of Pseudo Atomic Orbital Method)

Total energy Pseudopotentials Basis functions

Implementation: Total energy (1) The total energy is given by the sum of six terms, and a proper integration scheme for each term is applied to accurately evaluate the total energy.

Kinetic energy

Coulomb energy with external potential

Hartree energy

Exchange-correlation energy

Core-core Coulomb energy TO and H.Kino, PRB 72, 045121 (2005)

The reorganization of Coulomb energies gives three new energy terms.

The neutral atom energy

Difference charge Hartree energy

Screened core-core repulsion energy

Neutral atom potential Difference charge

Implementation: Total energy (2)

Short range and separable to two-center integrals

Long range but minor contribution

Short range and two-center integrals

So, the total energy is given by

}

}

Each term is evaluated by using a different numerical grid with consideration on accuracy and efficiency.

Spherical coordinate in momentum space

Real space regular mesh

Real space fine mesh

Implementation: Total energy (3)

Two center integrals Fourier-transformation of basis functions

e.g., overlap integral Integrals for angular parts are analytically performed. Thus, we only have to perform one-dimensional integrals along the radial direction.

Cutoff energy for regular mesh

The two energy components Eδee + Exc are calculated on real space regular mesh. The mesh fineness is determined by plane-wave cutoff energies.

The cutoff energy can be related to the mesh fineness by the following eqs.

Forces

Easy calc.

See the left

Forces are always analytic at any grid fineness and at zero temperature, even if numerical basis functions and numerical grids.

Total energy Pseudopotentials Basis functions

D. Vanderbilt, PRB 41, 7892 (1990).

The following non-local operator proposed by Vanderbilt guarantees that scattering properties are reproduced around multiple reference energies.

If the following generalized norm-conserving condition is fulfilled, the matrix B is Hermitian, resulting in that VNL is also Hermitian.

If Q=0, then B-B*=0

Norm-conserving Vanderbilt pseudopotential I. Morrion, D.M. Bylander, and L. Kleinman, PRB 47, 6728 (1993).

This is the norm-conserving PP used in OpenMX

Non-local potentials by Vanderbilt Let’s operate the non-local potential on a pseudized wave function:

Noting the following relations:

It turns out that the following Schrodinger eq. is hold.

Generalized norm-conserving conditions Qij

In the Vanderbilt pseudopotential, B is given by

Thus, we have

By integrating by parts, this can be transformed as

As well, the similar calculations can be performed for all electron wave functions.

・・・(1)

・・・(2)

By subtracting (2) from (1), we obtain a relation between B and Q.

Norm-conserving pseudopotential by MBK I. Morrion, D.M. Bylander, and L. Kleinman, PRB 47, 6728 (1993).

If Qij = 0, the non-local terms can be transformed to a diagonal form. The form is equivalent to that obtained from the Blochl expansion for TM norm-conserving pseudopotentials. Thus, common routines can be utilized for the MBK and TM pseudopotentials, resulting in easiness of the code development.

To satisfy Qij=0 , pseudofunctions are now given by

The coefficients {c} are determined by agreement of derivatives and Qij=0. Once a set of {c} is determined, χ is given by

Radial wave function of C 2s Pseudopotential for C 2s and –4/r

Red: All electron calculation Blue: Pseudized calculation

Pseudo-wave funtion and psedopotential of carbon atom

Optimization of pseudopotentials

1. Choice of valence electrons (semi-core included?) 2. Adjustment of cutoff radii by monitoring shape of

pseudopotentials 3. Adustment of the local potential 4. Generation of PCC charge

(i) Choice of parameters

(ii) Comparison of logarithm derivatives

If the logarithmic derivatives for PP agree well with those of the all electron potential, go to the step (iii), or return to the step (i).

(iii) Validation of quality of PP by performing a series of benchmark calculations. good

No good

No good

good

Good PP

Optimization of PP typically takes a half week per a week.

Comparison of logarithmic derivatives Logarithmic derivatives of s, p, d, f channels for Mn. The deviation between PP and all electron directly affects the band structure.

Comparison of band structure for fcc Mn

Total energy Pseudopotentials Basis functions

Primitive basis functions 1. Solve an atomic Kohn-Sham eq. under a confinement potential:

2. Construct the norm-conserving pseudopotentials.

3. Solve ground and excited states for the the peudopotential for each L-channel.

s-orbital of oxygen

In most cases, the accuracy and efficiency can be controlled by

Cutoff radius Number of orbitals PRB 67, 155108 (2003)

PRB 69, 195113 (2004)

Convergence with respect to basis functions

molecule bulk

The two parameters can be regarded as variational parameters.

Benchmark of primitive basis functions

Ground state calculations of dimer using primitive basis functions

All the successes and failures by the LDA are reproduced by the modest size of basis functions (DNP in most cases)

Variational optimization of basis functions

One-particle wave functions Contracted orbitals

The variation of E with respect to c with fixed a gives

Regarding c as dependent variables on a and assuming KS eq. is solved self-consistently with respect to c, we have

Ozaki, PRB 67, 155108 (2003)

→

Optimization of basis functions 1. Choose typical chemical environments

2. Optimize variationally the radial functions

3. Rotate a set of optimized orbitals within the subspace, and discard the redundant funtions

× ×

Database of optimized VPS and PAO

Public release of optimized and well tested VPS and PAO so that users can easily start their calculations.

Δ-factor The delta factor is defined as difference of total energy between Wien2k (FLAPW+LO) and a code under testing, which is shown as shaded region in figure below, where the volume is changed by plus and minus 6 % taken from the equilibrium V0 .

Lejaeghere et al., Critical Reviews in Solid State and Materials Sciences 39, 1-24 (2014).

Comparison of codes in terms of Δ-factor

http://molmod.ugent.be/deltacodesdft

How to choose basis functions: Si case(1) 1

2

3

4

5

6

7

8

Si7.0.pao Orbitals with lower eigenvalues in Si7.0.pao are taken into account step by step as the quality of basis set is improved.

Si7.0-s2p2d1 is enough to discuss structural properties. By comparing Si7.0-s3p2d2f1 with Si8.0-s3p2d2f1, it turns out that convergence is achieved at the cutoff of 7.0(a.u.).

With respect to band structure, one can confirm that Si7.0-s2p2d1 provides a nearly convent result.

While the convergent result is achieved by use of Si7.0-s3p2d2f1(Si7.0-s3p3d2f1), Si7.0-s2p2d1 is a balanced basis functions compromising accuracy and efficiency to perform a vast range of materials exploration.

How to choose basis functions: Si case(2)

Ongoing and planned developments

Speeding-up and low computational memory

Spectral calculations

Stress

Beyond GGA

Electronic transport

Molecular dynamics

O(N), O(N3), SCF, geometry optimization

Phonon dispersion, band unfolding, visible-ultraviolet, IR, Raman, XAS, NMR, ESR

Stress, local stress, local energy

Hybrid functionals, vdW-DF, mBJ, Machine-learning functionals, GW, RPA

Gate bias, eigenchannel, real space current, phonon scattering

Red: Ongoing Black: Planned

Constant μ-ESM, Blue moon, Neural network potentials

New functionalities in the next release

Opencore pseudopotentials for rare earth elements Band unfolding method Natural atomic and bond orbitals method Blue moon ensemble Constant μ effective screening medium (ESM) method Total energy decomposition method Voronoi volume Control of strength of spin-orbit coupling in OpenMX Constraint scheme for direction and magnitude of spin Crystal field analysis Stress Optical conductivity

The release is planned around the end of 2015 or the beginning of 2016.

Summary

OpenMX is a program package, supporting DFT within LDA, GGA, and plus U, under GNU-GPL.

The basic strategy to realize large-scale calculations relies on the use of pseudopotentials (PPs) and localized pseudoatomic orbital (PAO) basis functions.

The careful evaluation of the total energy and optimization of PPs and PAOs guarantee accurate and fast DFT calculations in a balanced way.

New functionalities are now under development.